D-A2lg U5 THEORY OF GYROTRON AMPLIFIERS IN A DISK OR A HELIX emILORDED HAVEGUIDECU) NAVAL SURFACE MEAPONS CENTER SILVERSPRING RD J Y CHOE ET AL. DEC 82 N5MC/TR-82-326

UNCLARSSIFIED SBI-RD-F588 M4 F/G 9/2 M

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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS+1963-A

- NSWC TR 82-526

PLASMA PHYSICS PUBLICATION NO. 82-12

THEORY OF GYROTRON AMPLIFIERS IN A DISK

OR A HELIX LOADED WAVEGUIDE

BY JOON Y. CHOE,HAN S. UHM

RESEARCH AND TECHNOLOGY DEPARTMENT

DECEMBER 1982

Approved for public release, distribution unlimited.

-- MAR 1 81983

144

C. NAVAL SURFACE WEAPONS CENTER

LL/ Dahlgren, Virginia 22448 0 Silver Spring, Maryland 20910

8

683038 OO

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REPORT DOCUMENTATION PAGE BEo INSTRUCTIONSBEFORE COMPL.ETING FORMI. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

NSWC TR 82-526 D A /_ _ _ _ _

4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

THEORY OF GYROTRON AMPLIFIERS IN A DISK OR A

HELIX LOADED WAVEGUIDES. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(& I. CONTRACT OR GRANT NUMUER(s)

Joon Y. Choe and Han S. Uhm

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKAREA & WORK UNIT NUMBERSNaval Surface Weapons Center (Code R41) 6l52N; ZR0000; ZR0119;

White Oak 65;AA400

Silver Spring, MD 20910

11. CONTROLLING OFICE NAME AND ADDRESS 12. REPORT DATEDecember 198213. NUMBER OF PAGES

4614. MONITORING AGENCY NAME & AODRESS(If different ftom Controlling Office) IS. SECURITY CLASS. (of this report)

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16. DISTRIBUTION STATEMENT (of thie Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstraect entered In Block 20, it different from Report)

I8. SUPPLEMENTARY NOTES

SS19. KEY WORDS (Continue on reverse side if nocessary and Identify by block numeber)

Gyrotron AmplifiersDisk Loaded WaveguideHelix Loaded Waveguide

20. ABSTRACT (Continue an ,ovorao aid. If necesar'y and Identify by block ntmber)

The gyrotron amplifiers in two slow wave structures, the disk and the helixloaded waveguides, are investigated for wide band applications. For eachstructure, properties of the vacuum waveguide mode are examined in connectionwith the gyrotron application, and the gyrotron dispersion relation isobtained. For the disk loaded gyrotron, it is found that the group velocityof the waveguide mode can be easily varied by the disk parameters, and thegain and the bandwidth are at least comparable to those of the ordinary&Lrotrnn. In the helix waveguide configuration, the zyrotron utilizing the

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helix mode can be very broad in its bandwidth. Moreover, the hybrid modeoperation, using the helix mode as a carrier, shows possibility of fast wavewide band amplification by contouring waveguide.

.-°

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SECURITY CLASSIFICATION OFr""IS PAGE(whel Date Entered)

NSWC TR 82-526

1K

FOREWORD

The gyrotron amplifiers in two slow wave structures, the disk and the helix

loaded waveguides, are investigated for wide band applications. For each

- structure, properties of the vacuum waveguide mode are examined in connection

Swith the gyrotron application, and the gyrotron dispersion relation is obtained.

* For the disk loaded gyrotron, it is found that the group velocity of the waveguide

mode can be easily varied by the disk parameters, and the gain and the bandwidth

* are at least comparable-to those of the ordinary gyrotron. In the helix waveguide

configuration, the gyrotron utilizing the helix mode can be very broad in its band-

width. Moreover, the hybrid mode operation, using the helix mode as a carrier,

shows possibility of fast wave wide band amplification by contouring waveguide.

Approved by:

IRA M. BLATSTEIN, HeadRadiation Division

rb 1

-" 'i

FOREWOR

NSWC TR 82-526

CONTENTS

Section Page

I INTRODUCTION ........... ......................... 1

II DISK LOADED GYROTRON .......... ..................... 3

III HELIX LOADED GYROTRON .......... .................... 9

IV CONCLUSION .......................... 17

V ACKNOWLEDGEMENT .......... ...................... .18

BIBLIOGRAPHY ......................... 29

OF

0 .

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ILLUSTRATIONS

Figure Page

1 SYSTEM CONFIGURATION OF A DISK LOADED WAVEGUIDE .. ......... ... 19

2(a) DEPENDENCE OF THE DISPERSION ON 7HE DISK GEOMETRY (a/b) ..... ... 20

2(b) DEPENDENCE OF THE DISPERSION ON THE PERIOD (L) .. ......... . 21

3 AN EXAMPLE OF THE DISK LOADED GYROTRON AMPLIFIER .. ........ . 22

4 SYSTEM CONFIGURATION OF A SHEATH HELIX LOADED WAVEGUIDE ..... ... 23

5(a) PLOTS OF DISPERSION CURVES FOR THE HELIX MODES WITH = ,

R0/Rh = 1.43 FOR SEVERAL VALUES OF Z ..... .............. . 24

5(b) PLOTS OF DISPERSION CURVES FOR THE HELIX MODES WITH = Tr/6,2k. = 2 FOR SEVER.. vaLUES OF R/Rh................... 25

6 PLOTS OF DISPERSION CURVES FOR THE n = 1 HYBRID MODES WITH/ = i/6, Rc/Rh = 1.4 FOR SEVERAL VALUES OF Z .. .......... 27

7 PLOTS OF THE GAIN VERSUS THE FREQUENCY FOR THE HELIX MODE WITH

A = 2%, Rc/Rh = 1.1, R0 = Rh - rL FOR SEVERAL VALUES OF Z . .... 7

8 PLOTS OF THE GAIN VERSUS THE FREQUENCY FOR THE n = 1 HYBRID WAVEWITH Z = 0, R0

= Rh - rL' 4) -600, R/cRh = 1.5, Rh w cc =

2.25 FOR SEVERAL VALUES OF A ....... .................. . 28

[."K

~iii/iv

NSWC TR 82-526

I. INTRODUCTION

The gyrotron (Flyagin et al 1977, Hirschfield and Granatstein 1977)

is a high power microwave tube that utilizes the fast wave coupling of the

waveguide mode with the cyclotron frequency upshifted beam mode. One of the

I.i shortcomings of the gyrotron, however, is its narrow bandwidth, primarily due

to the fast variation of the group velocity of the waveguide mode in a smooth

wall configuration. Therefore, it is necessary to slow down the wave in order

to broaden the bandwidth. In this regard, we have to re-examine various slow wave

. structures that have been widely used in conventional traveling wave tubes

* (Pierce 1950), in the context of the fast wave interaction of the gyrotron.

In this paper we will examine the gyrotron in two of such slow wave structures;

the periodic disk loaded gyrotron and the sheath helix loaded gyrotron. Another

slow wave structure, the dielectric loaded gyrotron, has been exhaustively

-' investigated already (Choe et al 1981).

Previous analyses of these two configurations exist (Chu and Hansen 1947,

* Walkinshaw 1948, Pierce 1950, Hutter 1950, Sensiper 1955), but these are limited

to the slow wave applications to the TWT. In the periodic disk configuration

• the transverse magnetic (TM) mode is analyzed, and for the sheath helix waveguide

"" most of them have ignored important role of the outer conducting wall. Here we

will investigate the transverse electric (TE) mode suitable for the gyrotron

interaction for the periodic disk geometry, and include the conducting wall in our

analysis for the helix geometry to utilize the location of the wall as a useful

parameter. The periodic disk loaded gyrotron will be discussed in Sec. II, and

the sheath helix loaded gyrotron in Sec. III.

In view of the role that the vacuum waveguide mode plays in performance of

the tubes, we will first examine the dispersion characteristics of the beam-free

waveguide mode for each structure (in Subsection A). This will be followed by the

i1

NSWC TR 82-526

derivation of the gyrotron dispersion relation with the electron beam in

Subsection B. For each structure, the Subsection B will include some numerical

examples of the gyrotron gain. We emphasize that the objective of this paper is

to examine the dispersion characteristics of the waveguide and to derive the

-- gyrotron dispersion relation for two slow wave structures. Therefore, the detailed

* parametric optimization with respect to the gain and the bandwidth will be

"- deferred for later works, only giving few typical examples.

2

'0"

2

NSWC TR 82-526

II. DISK LOADED GYROTRON

We consider a cylindrical waveguide loaded with periodic disks as shown in

' Fig. 1. The thin disks extend radially from the outer conductor at r = b to

r = a as shown, with the axial period given by L. The cylindrical coordinates

(r,e,z) are employed.

This configuration has been studied before in the context of the slow wave

TWT interaction (Chu and Hansen 1947, Walkinshaw 1948). Thus these analyses have

been limited to the TM modes. In order to utilize the same configuration for

the fast wave interaction of the gyrotron, therefore, we must examine the TE modes.

Otherwise the present analysis is similar to the previous ones. Emphases are,

hence, on the differences between TM and TE modes. In Subsection A, the TE

dispersion relation for the vacuum waveguide mode is obtained and discussed. The

gyrotron dispersion relation with the electron beam is derived and an example of

the gyrotron gain from this dispersion relation will be given in Subsection B.

A. Vacuum Waveguide Dispersion Relation

In this subsection we will derive and discuss the vacuum waveguide

dispersion relation for the periodic disks shown in Fig. 1. We limit our attention

to the azimuthally symmetric (i.e. 3/36 = 0, 0 = ) TE modes. Moreover, we will

employ an approximate scheme in order to obtain an analytical result.

The procedure to obtain the approximate dispersion relation is given

elsewhere (Choe and Uhm 1982), and only the outline will be given here. First,

the domain is divided into two separate regions (I for r <a, II for a ' r < b,

see Fig. 1), and the electromagnetic fields for TE mode are expressed in the separate

eigenfunction expansion for each region. The eigenfunctions are determined by the

boundary conditions on the metalic boundaries and the axial periodic conditions

4(i.e. Floquet theorem, Slater 1950). Then, the dispersion relation results in by

matching the fields at the mouth of the disks (i.e. at r = a), where an approximate

method is used to simplify the dispersion relation.

3

NSWC TR 82-526

7 The fields for each region (I and II) can be expressed as the sum of the

space harmonic (n) waves in region I, and of the standing waves with the nodal

number m for region II. Namely,

B zI(rz) = C nJ o(pn r) exp (ik nz), (1)

BzII(r z) = Dm [N(b ) J0 (qmr)" Jl(qmb) No (qmr)]

m=l* (1)'

x sin (KmZ),

where

k . /2 2 2 2

S2n/L, Pn- (2)

S2 2.c2 2K m -mr/L, 26 2 (2)'

Here w and k refer to the frequency and the axial wavenumber, respectively, and

*J and Nt to the first and second kind of Bessel functions of order £. Other

TE components (Eo, Br) are given in terms of B . Here E and B denote the electricr z

and magnetic fields. We note that the expression for Ez in TM mode is similar

to eq. (1), with one notable exception. For TE mode the first term in region II

is with m = 1, while it is with m = 0 for TM mode. The usual periodicity of the

dispersion relation in the axial wavenumber k with the period of 2w/L, and the

symmetry with respect to k are evident from eq. (1), even without the field

matching considerations. Thus it is sufficient to obtain the dispersion relation

for 0 < k </L.

Since an analytical dispersion relation is sought in this paper, we need an

approximation scheme for matching fields at the mouth (r = a). Instead of

matching fields for all z, we only match the average fields over the one axial

40"

NSWC TR 82-526

period L. That is, we demand that <qI> = <Ii >, where p stands for the fields

L Land <*> = Jdz 4(r = a,z). Furthermore, we assume that the axial wavenumber is

small, that is,

kL << . (3)

Then it can be shown (Choe and IUhm 1982) that the dominant term is for n = 0 in

region I and for m = 1 for region II, and that the average fields are matched

provided eq. (3) is valid. This can be compared with TM mode where, with the

same approximation scheme (eq. (3)), the n = 0 term in region I couples with the

m = 0 term in region II. Within this truncation method, the dispersion relation

*f for the TE mode is given by

D = j (p 0 a) - *l(qlb,qla) = 0 , (4)

where

PJ (x) J Jl(X)/[Xi 0(X)],

Ni(Yb)Jl(Ya) i(Yb) NI(y a ) (5)i(Yb'Ya- Ya [Ni(Yb)JO(Ya) - Ji(Yb)NO(Ya) ]

We recall that the corresponding TM dispersion relation is given byTM 2 2

DTM = j(pOa) - 00 (q0b,q0a) = 0. Here pn and qm are previously defined in

eq. (2). The difference in the dominant term in region II (m = 1 for TE,

m = 0 for TM) is reflected in the dispersion relation (4 for TE, 0 for TM).

Careful examination of the dispersion relation reveals several characteristics

of the disk loaded waveguide mode. For given k, the frequency w is higher than

that for the smooth wall (i.e. a = b). The reverse is true for the TM mode.

5

*i NSWC TR 82-526

* Also the frequency increases for given k as the period L decreases, and as the

ratio a/b decreases. This can be compared with the TM mode, where w is

independent of L and increases as a approaches b (smooth wall). The lower

* cutoff frequency w. increases as L increases and as a/b decreases. For TM

mode, wco is independent of both L and a/b. These dependence of the frequency

on the period L and the ratio a/b are further illustrated in Fig. 2. In

Fig. 2(a), the TE mode dispersion curves for several disk dimensions a/b are

plotted at given period (irb/L = 10.0). It is evident from the graph the

cutoff frequency w co at k = 0 is decreasing function of a/b. At the same time,

* if we extroplate our analysis to the limit at k = w/L, the frequencies for all

*a/b converge to that for the smooth wall. This pivot-like phenomenon occurs

at k = 0 for the TM modes. Obviously the group velocity of the wave decreases

as the ratio a/b decreases. The dependence on the period L is shown in Fig. 2(b).

Here the dispersion curves are drawn for several L at given disk dimension

(a/b = 0.5). Also shown is the dispersion curve for the smooth wall (broken

curve) for reference. Evidently for given k, w is an increasing function of L.

The graph also reveals that the group velocity of the wave de' reases as L

increases. We conclude that the group velocity can be easily adjusted by

variation of the disk parameters, a/b and L.

40B. Gyrotron Dispersion Relation

In this subsection we will derive the dispersion relation for the disk

loaded gyrotron in the presence of the electron beam. It is assumed that the

thin hollow electron beam is located at r = RO(O < R0 < a), under the influence

of the strong axial magnetic field B0, and the configuration otherwise is the

same as in Fig. 1. Again we limit our attention to the azimuthally symmetric

0 (Z = 0) TE mode.

6

|

-r - ..

NSWC TR 82-526

The analysis is carried out within the framework of the linearized Maxwell-

r Vlasov system for the fields E and B, and for the beam electron distribution

function f. Following the scheme in previous subsection, the perturbed fields

are for ihose with the fundamental space harmonic (n = 0) and with the first

. nodal number (m = 1). Namely, in the vacuum region,

AIJ0 (p0 r) 0 < r < R0 ,iz

Bz A2Jo(p0r) + A3N0 (p0r) R0 i r < a , (6)z 2

A4J0 (qlr) + A5N0 (qlr), a < r < b

The coefficients A's are to be determined from the boundary conditions, the field

matching condition at r = a (Subsection A), and the jump condition on B at thez

beam location R0. The jump condition on B at R0, in turn, is obtained from the

moment equation with the purturbed distribution function. In the present analysis

we assume that the equilibrium distribution function f0 is Lorentzian in the axial

momentum Pz. Namely,

fa p / p - Z)2 + A2 1] (7)

where A refers to the axial momentum spread ratio. We further designate the

total electron energy and the average axial (transverse) velocity by ymc2 and

caz (c8,), respectively. The jump condition at R0, or equivalently impedance

matching across the beam (Choe et al 1981), provides the dispersion relation for4

the gyrotron.

B BN v 12 c2.B - B 2'R r 312(8wh D 0 WB + icfkl yA/Y3

where

7

NSWC TR 82-526

'-B N 2B1 BD _ x 2 Jl1(xO) [Jl(xO) B2 - Nl(xO) B1 ]

:: ~ ~B = N0 (xa) [*N(Xa) "- iY'a222

l t i h e t lig n =e an a te e r c

x 2Ra 2 2 ba2

eqs. ,N(X) d (X)/ x NO(x)

W WB = kC z + c/ ' Y z = I - Bz 2,

and vc u w /mg is the non-relatrelatic cyclotron frequency and v = 0).r-i is the Budker parameter. Here N is the total numnber of electrn pe nt xa

i. length, c is the velocity of light, and (-e) and m are the electronic charge

[t and mass. The quantities, p., qlp *J, 01, and A are previously defined in

Seqs. (2), (5) and (7). Note that BI1 a DTE so that the equation (8) recovers

i" the vacuum waveguide dispersion relation (4) when the beam is absent (v = 0).

The dispersion relation (8) can be used to investigate gain and bandwidth of

r the disk loaded gyrotron amplifier for a broad range of physical parameters.

Here we will give only one example in Fig. 3. At the given system parameters

(v = 0.002, 1 = 0.4, Oz = 0.2, a/b = 0.3, awc/c = 2.8, L wc/C = 3.2,

SR 0 /C = 2.0), the normalized gain (-100 k. c/W ) versus the normalized0Uc 1 cfrequency (w/w c) is plotted for several values of the axial velocity spread

* (A). Even without the benefit of the parametric optimization, both the gain

and the bandwidth for this disk loaded gyrotron are at least comparable to those

for the smooth wall gyrotron. We will leave the parametric optimization as

future works.8

NSWC TR 82-526

III. HELIX LOADED GYROTRON

As illustrated in Fig. 4, the system configuration consists of a sheath

helix with radius Rh located inside a cylindrical conductor at Rc. We assume

that conducting wires in the helix are very fine in texture, thereby replacing

the actual helix by a helically conducting sheet. This sheet is perfectly

conducting in a helical direction making a pitch angle 0 with a plane normal

to the axis of the system, as shown in Fig. 4. The effect of the helix is

then restricting the current over the sheet in * direction, while allowingthe fields penetration to Rh>r>Rc. Again the cylindrical coordinates are

used.

Although this helix structure has been investigated before (Pierce 1950,

Sensiper 1955, Berezin et al 1960), these studies either neglect the role of

the outer conductor, thereby restricting to slow wave propagation only, or

concentrate only on the azimuthally symmetric (Z=O) mode. We, therefore,

investigate the dispersion characteristics for arbitrary azimuthal mode

number (Z O) including the outer conductor. As before, the vacuum waveguide

dispersion relation is obtained and analyzed in subsection A. Two interesting

limits, namely as Rc - - and as Rc - Rh, are examined. The former recovers

previous results and the latter yields clear pictures of the mode identifi-

cations. The gyrotron dispersion relation with the electron beam will be

derived and some numerical examples will be given in subsection B.

A. Vacuum Waveguide Dispersion Relation

Since the detailed procedure of the vacuum dispersion relation

is given elsewhere (Uhm and Choe 1982a), only the brief outline will be given

9

NSWC TR 82-526

here. In anticipation of the presence of the hybrid modes, we express both

Ez and Bz as a general solution of the wave equation for the Fourier component

t (azimuthal mode number), k (axial wavenumber) and w (frequency). Then the

boundary conditions on the wall and on the helix yield the desired dispersion

relation.

The primary fields E and B are given byz z

A1 J 1 (pr) 0 < T < R

(10)

AE LNt(PRc) J (pr) - Je(PRc) N1 (pr) R < r < Rc

A1 B J,(pr), 0 < r < RhB ;

z

A B N(pR) J,(pr) -J(pRc) N~pr)l, Rh< r<R2 t~ C c I R

where

2. w2 /C2 k2J2p -_ W/c2 - 2 , (11)

and the prime (') denotes the derivative of Bessel functions. Other components

41 of the fields are given in terms of Ez and B . The boundary conditions on the

sheath helix (r=R h) are

1hE e E " (12)E x e=R E x e 0

* 10

7,77 47 1, 17

NSWC TR 82-526

where eo is the unit vector in the helix direction *, and Rh! refers to the

evaluation at just inside (-) or just outside (+) the helix surface. The

dispersion relation for the vacuum waveguide mode is obtained by combining

eqs. (10) and (12). The resultant dispersion relation is given by

W 2 2 2D =2 N JRJ(Xc)Je(Xh) + X t tJ(Xc)JZ(Xh)f 0 (13)

c

where

f fN/fD, tZ = tano - f-kRh/Xh 2 ,

f N = JE(Xc)NZ(Xh) - NL(Xc)JZ(Xh)'

fD = Ji(Xc)Ni(Xh) - Ni(Xc)Ji(Xh) , (14)

c2

2 2

and p is defined in eq. (11). Note that the sign of p2 determines whether the

wave is fast (w>ck, p 2>0) or slow (w< ck, p2< 0).

In the limit R -*, the dispersion relation (13) is further simplified asE ~ c'

W 2 p2tz2 Ji(Xh) N(Xh) (15)c2;..". J-(Xh) N-(Xh

which has solutions only when p < 0. This means that when the outer conductor

is removed, only the slow wave (w < ck) can propagate. The dispersion relation

(15) with Bessel functions J and N explicitly given by modified functions I and K

411

-!

NSWC TR 82-526

can be found in previous literatures (Pierce 1950, Hutter 1950, Sensiper 1955).

*More interesting features can be found in the other limit. Namely, on the

limiting case when the outer conductor approaches to the helix (i.e. R R

eq. (13) is reduced to

(2J (X() Jz(X) - [k sin + (/RC) cos] 2 0. (16)

C C

From eq. (16), we identify three distinctive solutions; the transverse electric (TE)

modeW2 V~ 2 'ts 2

. 2 -(17)

the transverse magnetic (TM) mode

-•)2 2 82s-- k = -,(18)

c RC

and the helix mode

= ± [ck sine + Z(c/Rc) cosJ] (19)

where cY,, and 8~ are the Sth roots of Jo (ct) = 0 and

Jt(ts) = 0, respectively.

It is remarkable to note from eq. (19) that the helix mode for R Rh is

a straight line in the (w,k) space. Thus, when the helix mode with Rh/R 1

is coupled with the electron beam mode (w B = kcz + w c/y, eq. (9)) for the

gyrotron, a super wide band amplifier can be constructed by a choice of

appropriate beam parameters satisfying

. = sine, R =ycZ cos /c. (20)2 C C

12

6

.1 NSWC TR 82-526p..

In general case where R /Rh 1, there exist two distinguishing kinds of

modes; (a) the hybrid waves mainly consisting of a combination of the TE and

TM modes and (b) the helix mode. The dispersion relation (14) is numerically

solved for w at given k. Shown in Fig. (5) are plots of the normalized

frequency wR /c versus the normalized wavenumber kRc obtained from eq. (14)

for.the.helix mode, I = 7r/6, (a) Rc/Rh = 1.43 and several values of t and (b)

. = 2 and several values of Rc/Rh. We note from Fig. 5(a) that for the t + 0

helix mode, the dispersion curve for the slow wave region (w < ck) is continu-

ous.ly connected to that for a portion of fast wave region (w > ck). On the

other hand, the dispersion curve for the Z = 0 helix mode consists entirely

of the slow wave region. As mentioned above, the dispersion curves of the

helix mode in Fig. 5(b) approach to the straight lines described by eq. (19)

as the parameter Rc/Rh decreases to unity.

As expected, there are infinite number of the hybrid waves which are

identified by the radial node number n. All of these hybrid modes are fast

waves (w > ck). Figure 6 is the plot of the frequency versus the wavenumber

obtained from eq. (14) for the n = 1 hybrid waves, = i/6, Rc/Rh = 1.43,

and several values of Z. Note that the dispersion curves for the t 0 hybrid

waves is not symmetric about k = 0 line. Additional information on the vacuum

dispersion properties are given in our previous paper (Uhm and Choe 1982a).

B. Gyrotron Dispersion Relation

The dispersion relation for the helix loaded gyrotron in the

presence of the electron beam will be derived in this subsection. The thin

hollow electron beam, located at r = R0 (0 < R0 < Rh), is assumed to be

embedded in the strong axial magnetic field B0. The system configuration

is otherwise same as given in Fig. 4.

13

NSWC TR 82-526

The detailed derivation of the dispersion is given elsewhere (Uhm and

Choe 1982b). The general procedure is similar to that given in Sec. TI B.

:Within the framework of the linearized Maxwell-Vlasov system, the perturbed

fields in the vacuum region are given as general solutions of the wave

equation similar to eq. (10), and the boundary condition on the helix (eq. (12))

and the jump condition on Bz across the beam location yields the dispersion

relation. The jump condition is given by the moment equation of the perturbed

distribution function, which is computed from the equilibrium distribution

function given by eq. (7).

Then we finally obtain the dispersion relation for the Fourier component

Z (azimuthal), k (axial wavenumber), and w (frequency) in the helix loaded

waveguide.

2 2'; BN Val c

B = N _ 8(21)BD 2y R0 . w w + icl' WBzya/Yz

where

B 2B1 9 BD n X 8JT l(X t_ 1(X)B - 1-

21 Rh2Jt5Xc)J-(Xh)fD 2 hZZXJZX~N

B2 = Rh2 Jz(X )Ni(Xh)fD + Xh2t 2 Ni(Xc)JZ(Xh)fN (22)

- kR hXht/-fDfl

x02 = 2

0 14

I-I

NSWC TR 82-526

Here the quantities, Xh, Xc, fD' fN' tZ and p2 are defined in eq-. (14) and (11).

The beam mode wB is defined in eq. (9) and the axial velocity spread ratio

appears in eq. C7). We note that in the limit of v -) 0, the dispersion

relation (21) recovers the vacuum waveguide dispersion equation (13) (i.e.

B1 D). The dispersion relation in eq. (21) can be used to investigate gain

and the bandwidth of the helix loaded gyrotron amplifier for a broad range of

physical parameters.

In the rema.nder of this subsection we show several numerical examples of

the gyrotron gain (ki) versus the frequency with assumed beam parameters, v=0.002,

= 0.4, $z = 0.2. In order to illustrate the broad band amplification in a

helix loaded waveguide, Fig. 7 shows a plot of the normalized gain (-100kic/Wc)1 C

versus the normalized frequency w/w obtained from eq. (21) for the helix mode,c

A = 2%, RC/R = 1.1, R0 = Rh - rL, and optimum values of the parameters

(R hwc/cp) for each azimuthal harmonic number t. The optimum values of the

parameters (Rh Wc/c,O) are given by (1.04, 10.60) for Z = 1, (2.09, lI.0*) for

Z = 2, (3.15,. 11.00) for t = 3 and (4.21, 11.00) for Z 4. The normalized

electron Larmor radius is rLwc/d = 0.447. As expected in previous subsection,

utilizing the helix mode for Rh/RC = 1, it is shown that a super wide band

gyrotron amplifier is attainable for 8z = sine and Rc = YcL coso /Wc (eq. (20)).

For example, for t = 2 and A = 2% in Fig. 7, the amplifier bandwidth is more

K; than 60%. Here, the bandwidth is defined by the full width of the real

frequency, at which the linear gain drops to exp (-1/2) of its maximum value,

normalized by the mean frequency. When the axial velocity spread A is increased,

the gain and bandwidth are considerably reduced. However, for t = 2, the

bandwidth of the gyrotron amplifier is found to be still more than 40% for

A = 4%. We can conclude, therefore, the helix mode amplifier is a very

effective means to amplify a broad band microwave signal.

15

.. .

7 7, °7 17 .. . . . 7 . . .. . . .. . ... -777 . 7 7

NSWC TR 82-526

In addition to the helix mode, the fast hybrid mode can be used for the

gyrotron amplifier. As a typical example of the hybrid wave utilization,

Fig. 8 presents plots of the gain versus the frequency for - = 0, n = 1,

R0 = Rh - rL, and the geometric parameters Rc/Rh = 1.5, Rhw c/d = 2.25

and = - 60°,which satisfy the grazing condition. Also shown in

the horizontal line is k.c/wc = w/wc - 1/y. Although the vacuum waveguide mode

for (Z,n) = (0.1) is orginated from the TM mode, the maximum gain in Fig. 8 is

very similar to that for the ordinary gyrotron where the beam mode couples

with the TE mode. This strongly indicates that the hybrid wave for (Z,s) =

(0,1) TE and TM modes in eqs. (17) and (18). The fast wave wide band

amplification (Lau and Chu 1981) in a helix loaded waveguide is currently under

investigation by authors, utilizing these hybrid waves as an amplification

mechanism and the helix mode as a signal carrier. The general behavior of the

gyrotron gain remains the same for the hybrid modes with Z $ 0.

16

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IV. CONCLUSION

In this paper we have investigated the gyrotron amplifiers in a disk

and a helix loaded waveguides. For each slow wave structure, the

vacuum waveguide mode is derived and analyzed, and the gyrotron dispersion

relation is obtained.

The disk loaded waveguide is discussed in Sec. II. The dispersion

*- characteristics of the azimuthally symmetric TE mode are examined and compared

with those of the TM mode in Sec. IIA. It has been found that the low cutoff

frequency of the disk loaded TE mode is higher than that for the smooth wall,

and that the group velocity of the wave can be easily adjusted by disk parameters.

In Sec.IIB, the gyrotron dispersion relation in the disk configuration is obtained

and a numerical example of the gain is given. The gain and the bandwidth of the

disk loaded gyrotron is at least comparable to those for the smooth wall gyrotron.

In Sec. III, the helix loaded gyrotron is examined. The analysis of the

vacuum waveguide mode reveals that the helix configuration supports two

distinctive modes; the fast wave hybrid modes and the helix mode. In particular,

when the outer conductor is very close to the helix, the helix mode becomes a

straight line in the dispersion (w,k) space. From Sec. IIB, it is shown that,

by proper choice of the helix parameters, the bandwidth of the gyrotron utilizing

the helix mode can be very wide, in excess of 40% even for 4% of the velocity

spread. Also, by coupling with the hybrid mode and using the helix wave as a

signal carrier, the hybrid mode gyrotron can be used as one of fast wave wide

band schemes.

171

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V. ACKNOWLEDGEMENT

* This research was supported by the Independent Research Fund at the

Naval Surface Weapons Center.

*1

18

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CONDUCTOR

I. I# r

II

0 L 21 31

FIGURE 1. SYSTEM CONFIGURATION OF A DISK LOADED WAVEGUIDE

19

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10.0 ab= .

Coo

-b

5.0-

0.0 5.0 10.0kb

FIGURE 2A. DEPENDEWJE OF THE DISPERSION ON THE DISK GEOMETRY (A/B)

20

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TEO,

15.0

10.014.

10.0 '

bk

44.

5.

21

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TE01 GYROTRON

a/b =0.3

10.0-co/-3.

A0.00

5.00

0.01 11 1 1 1 1 11 1 1 1 1 1 1 10.90 0.95 1.00 1.05 1.10

COIWCo

FIGURE 3. AN EXAMPLE OF THE DISK LOADED GYROTRON AMPLIFIER. THE NORMALIZED GAIN(-100 kic/c%) FOR SEVERAL VALUES OF THE AXIAL VELOCITY SPREAD (A) IS PLOTTED

* AGAINST THE NORMALIZED FREQUENCY (w~wd)

22

. . . .. . . . . .. . , . . . . . . . . . .. .. . . . . .. . . . . . - . . -

NSWC TR 82-526

b~

r,

HEI

r

CONDUCTING WALLy

FIGURE4. SYSTEM CONFIGURATION OF A SHEATH HELIX LOADED AVEGUIDE

I/

4.-

|-.

!eo

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FIUR S. LOS F ISERIO CRVS (ORTH HELIX MODE H~=11.R/h=14

FOR SEVERAL VALUES OF Q

-4

24

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(2.05

(b4

NSWC TR 82-426

c kR 5 /

-55

FIGURES6. PLOTS OF DISPERSION CURVES FOR THE n-1 HYBRID MODES WITH 0-,r/6, RC/Rh - 1.4FOR SEVERAL VALUES OF 2

62

! II - _ .- - - _ I q

NSWC TR 82-526

HELIX MODE, A = 0.02,

10- Rc/Rh= 1.1, Ro = Rh - rL

b5.-

C.,

1=4

0

1.0 1.5 2.0oJoc

FIGURE 7. PLOTS OF THE GAIN VERSUS THE FREQUENCY FOR THE HELIX MODE WITH ' 2%,

Rc/Rh 1.1, R0 Rh - rL FOR SEVERAL VALUES OF R

27

NSWC TR 82-526

HYBRID WAVE, I Or0R=R Rh-rL =-600

RC/R h =1.5, RhwcO /c2.25r10GRZN

C0.0

000

0.90 0.95 1.00

0 0.5kbC/Wc

FIGURE&8 PLOTS OF THE GAIN VERSUS THE FREQUENCY FOR THE n -I1 HYBRID WAVE WITHQ - 0, R0 Rh - rL, 0 600, ReIR 1.5. Rh w0in2. 25 FOR SEVERAL VALUES OFA

28

- .. . .1

NSWC TR 82-526

BIBLIOGRAPHY

Berezin, A. K., Zeidlits, P. M., Nekhrshevich, A. M., and Silenok, G. A.,

1960, Soviet Phys. Tech. Phys., 4, 730.

Choe, J. Y., and Uhm, H. S., 1982, IEEE Int'l. Conf. on Plasma Science,

Conf. Rec., 62.

Choe, J. Y., Uhm, H. S., and Ahn, S., 1981, J. Appl. Phys., 52, 4508; and

g references therein.

Chu, E. L., and Hansen, W. W., 1947, J. Appl. Phys., 18, 996.

* Flyagin, V. A., Gaponov, A. V., Petelin, M. I., and Yulpatov, V. K., 1977,

IEEE Microwave Theory Tech., MTT-25, 514; and references therein.

Hirschfield, J. L., and Granatstein, V. L., 1977, IEEE Microwave Theory Tech.,

MTr-25, 528; and references therein.

Hutter, R. E. G., 1960, Beam and Wave Electronics in Microwave Tubes, Van Nostrand,

New York.

Lau, Y. Y., and Uhu, K. R., 1981, Int'l. J. Infrared and Millimeter Wave, 2, 415.

Pierce, J. R., 1950, Traveling-Wave Tubes, Van Nostrand, New York.

Sensiper, S., 195S, Proc. IRE, 43, 149.

Slater, J. C., 1950, Microwave Electronics, Van Nostrand, New York.

Uhm, H. S., and Choe, J. Y., 1982a, Properties of the electromagnetic wave

propagation in a helix loaded waveguide (submitted for publication); 1982b,

Gyrotron amplifier in a helix loaded waveguide (submitted for publication).

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